Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

We prove that a finite-dimensional irreducible Hopf algebra H in positive characteristic is semisimple if and only if it is commutative and semisimple if and only if the restricted Lie algebra P(H) of the primitives is a torus. This generalizes Hochschild\u27s theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel\u27s and Sweedler\u27s results on group schemes in the special but essential situation with a finiteness assumption added